3.7.89 \(\int \frac {A+B x}{x^{3/2} (a^2+2 a b x+b^2 x^2)} \, dx\)

Optimal. Leaf size=88 \[ -\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)} \]

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Rubi [A]  time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 78, 51, 63, 205} \begin {gather*} -\frac {3 A b-a B}{a^2 b \sqrt {x}}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

-((3*A*b - a*B)/(a^2*b*Sqrt[x])) + (A*b - a*B)/(a*b*Sqrt[x]*(a + b*x)) - ((3*A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x
])/Sqrt[a]])/(a^(5/2)*Sqrt[b])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx\\ &=\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {\left (-\frac {3 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{a b}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 59, normalized size = 0.67 \begin {gather*} \frac {(a+b x) (a B-3 A b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b x}{a}\right )+a (A b-a B)}{a^2 b \sqrt {x} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(a*(A*b - a*B) + (-3*A*b + a*B)*(a + b*x)*Hypergeometric2F1[-1/2, 1, 1/2, -((b*x)/a)])/(a^2*b*Sqrt[x]*(a + b*x
))

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IntegrateAlgebraic [A]  time = 0.10, size = 67, normalized size = 0.76 \begin {gather*} \frac {(a B-3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}+\frac {-2 a A+a B x-3 A b x}{a^2 \sqrt {x} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-2*a*A - 3*A*b*x + a*B*x)/(a^2*Sqrt[x]*(a + b*x)) + ((-3*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(5/
2)*Sqrt[b])

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fricas [A]  time = 0.44, size = 215, normalized size = 2.44 \begin {gather*} \left [\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{a^{3} b^{2} x^{2} + a^{4} b x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

[1/2*(((B*a*b - 3*A*b^2)*x^2 + (B*a^2 - 3*A*a*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a))
- 2*(2*A*a^2*b - (B*a^2*b - 3*A*a*b^2)*x)*sqrt(x))/(a^3*b^2*x^2 + a^4*b*x), -(((B*a*b - 3*A*b^2)*x^2 + (B*a^2
- 3*A*a*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (2*A*a^2*b - (B*a^2*b - 3*A*a*b^2)*x)*sqrt(x))/(a^3*b
^2*x^2 + a^4*b*x)]

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giac [A]  time = 0.16, size = 60, normalized size = 0.68 \begin {gather*} \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {B a x - 3 \, A b x - 2 \, A a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

(B*a - 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + (B*a*x - 3*A*b*x - 2*A*a)/((b*x^(3/2) + a*sqrt(x))
*a^2)

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maple [A]  time = 0.09, size = 87, normalized size = 0.99 \begin {gather*} -\frac {3 A b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {A b \sqrt {x}}{\left (b x +a \right ) a^{2}}+\frac {B \sqrt {x}}{\left (b x +a \right ) a}-\frac {2 A}{a^{2} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-1/a^2*x^(1/2)/(b*x+a)*A*b+1/a*x^(1/2)/(b*x+a)*B-3/a^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*A*b+1/a/(a*
b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*B-2*A/a^2/x^(1/2)

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maxima [A]  time = 1.41, size = 65, normalized size = 0.74 \begin {gather*} -\frac {2 \, A a - {\left (B a - 3 \, A b\right )} x}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} + \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

-(2*A*a - (B*a - 3*A*b)*x)/(a^2*b*x^(3/2) + a^3*sqrt(x)) + (B*a - 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b
)*a^2)

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mupad [B]  time = 1.17, size = 65, normalized size = 0.74 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {x\,\left (3\,A\,b-B\,a\right )}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b-B\,a\right )}{a^{5/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)),x)

[Out]

- ((2*A)/a + (x*(3*A*b - B*a))/a^2)/(a*x^(1/2) + b*x^(3/2)) - (atan((b^(1/2)*x^(1/2))/a^(1/2))*(3*A*b - B*a))/
(a^(5/2)*b^(1/2))

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sympy [A]  time = 19.07, size = 884, normalized size = 10.05 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {4 i A a^{\frac {3}{2}} b \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {6 i A \sqrt {a} b^{2} x \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 A a b \sqrt {x} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 A a b \sqrt {x} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {2 i B a^{\frac {3}{2}} b x \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {B a^{2} \sqrt {x} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {B a^{2} \sqrt {x} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {B a b x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {B a b x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(3/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a**
2, Eq(b, 0)), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2)))/b**2, Eq(a, 0)), (-4*I*A*a**(3/2)*b*sqrt(1/b)/(2*I*a**(7
/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) - 6*I*A*sqrt(a)*b**2*x*sqrt(1/b)/(2*I*a**(7/2)
*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) - 3*A*a*b*sqrt(x)*log(-I*sqrt(a)*sqrt(1/b) + sqrt
(x))/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) + 3*A*a*b*sqrt(x)*log(I*sqrt(a)
*sqrt(1/b) + sqrt(x))/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) - 3*A*b**2*x**
(3/2)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(
1/b)) + 3*A*b**2*x**(3/2)*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*
b**2*x**(3/2)*sqrt(1/b)) + 2*I*B*a**(3/2)*b*x*sqrt(1/b)/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*
x**(3/2)*sqrt(1/b)) + B*a**2*sqrt(x)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2
*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) - B*a**2*sqrt(x)*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I*a**(7/2)*b*sqrt(
x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) + B*a*b*x**(3/2)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(2*I
*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)) - B*a*b*x**(3/2)*log(I*sqrt(a)*sqrt(1/b)
+ sqrt(x))/(2*I*a**(7/2)*b*sqrt(x)*sqrt(1/b) + 2*I*a**(5/2)*b**2*x**(3/2)*sqrt(1/b)), True))

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